The level-set flow on a manifold. (English) Zbl 0827.53014

Ilmanen, Tom

Greene, Robert (ed.) et al., Differential geometry. Part 1: Partial differential equations on manifolds. Proceedings of a summer research institute, held at the University of California, Los Angeles, CA, USA, July 8-28, 1990. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 54, Part 1, 193-204 (1993).

Let \((M,g)\) be a Riemannian manifold and let \(f : M \to \mathbb{R}\) be a continuous function which defines an initial hypersurface \(\Gamma_0\) by \(\Gamma_0 = f^{-1} (0)\). Consider the equation \[ \begin{cases} {\partial \over \partial t} u - \left (g - {Du \otimes Du \over |Du|^2} \right) : D^2 u = 0 & \text{on } M \times [0,T) \\ u = f & \text{on } M \times \{t = 0\}. \end{cases} \] One says that each level set of \(u\) evolves according to its mean curvature. In the paper under review, the author describes various known techniques to study the above equation and announces the results of his paper which are published in [Indiana Univ. J. Math. 41, No. 3, 671-705 (1992; Zbl 0759.53035)]. Moreover, the author gives an alternative characterization of the level-set flow.
For the entire collection see [Zbl 0773.00022].

Reviewer: M.Hotloś (Wrocław)

MSC:

53B20	Local Riemannian geometry
35K15	Initial value problems for second-order parabolic equations

Keywords:

parabolic PDE; evolution by mean curvature; level-set flow

Citations:

Zbl 0759.53035